Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x⁴+5x³+5x²−5x−6;f(3)

Verified step by step guidance

1. Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = x^{4} + 5x^{3} + 5x^{2} - 5x - 6\) and you want to find \(f(3)\).
2. Set up synthetic division by writing down the coefficients of the polynomial: \$1\( (for \)x^{4}\(), \)5\( (for \)x^{3}\(), \)5\( (for \)x^{2}\(), \)-5\( (for \)x\(), and \)-6$ (constant term).
3. Write the value \$3\( (the input for \)f(3)$) to the left of the synthetic division setup.
4. Perform synthetic division by bringing down the first coefficient, then multiply it by \$3$, add this result to the next coefficient, and repeat this process across all coefficients.
5. The final number you obtain after completing synthetic division is the remainder, which equals \(f(3)\) by the Remainder Theorem.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making it faster and less error-prone. This method helps find the quotient and remainder efficiently.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means evaluating the polynomial at x = c gives the remainder directly, which is useful for quickly finding function values or checking factors.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial function at a specific input x = c. Using synthetic division or direct substitution, you can find f(c), which represents the output of the polynomial for that input. This is essential for understanding function behavior and solving related problems.